StATS: Re-weighting the data (January 25, 2005).
A recent article
shows how a simple re-weighting of the data can lead to a fairer comparison between two groups. An expanded version of this paper is available on the Statistical Literacy website.
Wainer and Brown show data on a state by state basis for the National Assessment of Educational Progress (NAEP). Two states, Nebraska and New Jersey show interesting results. The average score for Nebraska is 277 and is only 271 for New Jersey. But interestingly enough, New Jersey outperforms Nebraska among whites (283 vs 281), blacks (242 vs 236) and other non-white (260 vs 259).
This is an example of Simpson's paradox. New Jersey has much different demographics than Nebraska. In New Jersey 66% of the population is white, 15% black, and 19% other non-white. In Nebraska, 87% of the population is white, 5% is black, and 8% is other non-white. It is this differing demographic mix that causes the paradox.
The average score for each state is a weighted average. For Nebraska, the calculation is
281*0.87 + 236*0.05 + 259*0.08 = 277
and for New Jersey, the calculation is
283*0.66 + 242*0.15 + 260*0.19 = 272
Nebraska benefits because a higher weight (0.87) is placed on the race that scored highest in both states. What would happen to Nebraska's and New Jersey's scores if the demographic mix was changed to the overall percentages in the U.S. (69% white, 16% black, and 15% other non-white)?
Here are the re-weighted calculations for Nebraska
281*0.69 + 236*0.16 + 259*0.15 = 271
and New Jersey
283*0.69 + 242*0.16 + 260*0.15 = 273
This re-weighting to a common demographic risk is often used to make adjustments between two groups that have sharply differing mixes of age, gender, and/or racial characteristics.
Here are a few additional references about Simpson's paradox.
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